@article{Kaiser2024, author = {Kaiser, Tobias}, title = {Periods, power series, and integrated algebraic numbers}, series = {Mathematische Annalen}, volume = {390}, journal = {Mathematische Annalen}, number = {2}, publisher = {Springer Berlin Heidelberg}, address = {Berlin/Heidelberg}, issn = {0025-5831}, doi = {10.1007/s00208-024-02802-2}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2024040910285674698609}, pages = {2043 -- 2074}, year = {2024}, abstract = {Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the polynomial ring over the rationals and evaluate it at a rational number. We follow this path and close these algebraic power series under taking iterated antiderivatives and nearby algebraic and geometric operations. We obtain a system of rings of power series whose coefficients form a countable real closed field. Using techniques from o-minimality we are able to show that every period belongs to this field. In the setting of o-minimality we define exponential integrated algebraic numbers and show that exponential periods and the Euler constant is an exponential integrated algebraic number. Hence they are a good candiate for a natural number system extending the period ring and containing important mathematical constants.}, language = {en} }